Mainz were on target twice in the space of two minutes, thanks to two assists from Phillipp Mwene. First, his cross found Armindo Sieb who headed the ball into the net in the 13th minute.
Mwene then crossed the ball to Jonathan Burkardt, who also used his head to score Mainz's second in the 15th.
Augsburg pulled one back in the 25th after a free-kick was nudged back into the middle off the box and Keven Schlotterbeck was first to it, smashing in from close range.
The tide seemed to be turning when Mainz were down to 10 men after Nadiem Amiri was sent off for a second bookable offence in the 35th.
But Burkardt completed his brace early in the second half. Augsburg again pulled one back through Samuel Essende in the 57th but he was sent off just three minutes later for violent conduct, after kicking Dominik Kohr when none of them had the ball.
Augsburg thought they had a penalty in stoppage time after what seemed to be a foul by Silvan Widmer on Schlotterbeck, but the decision was overturned after a lengthy VAR review - to the home's team disappointment.
"It was an unbelievable game for the neutral spectators, there was a lot going on," Burkardt told broadcasters DAZN.
There was a lot of debate over the potential penalty to Augsburg, with Mainz coach Bo Henrikson saying it wasn't a penalty.
"Of course not, for me not a penalty," he said. "We gave everything, a great evening."
But Augsburg midfielder Elvis Rexhbecaj was furious and said it was "ridiculous" the penalty had been rescinded, but concluded it was a wrong decision after watching the video.
However, asked how Augsburg could lose the game after having 30 goal attempts, he said: "That's a good question. With 30 attempts, we should have had something. Very, very hard."
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